5. Extract the square root of 2025.

Ans. 45.

1st. Divide by the square number, 25, and we find the two factors, 25×81, as equivalent to the given number. Roots of these factors, 5X9=45, the answer.

2d. Divide by the square number, 100, and we have 20X100X100, X 10=45, as before.

we multiplied the power by 4, and we have 45 for the result.

6. What is the square root of 390625? Ans. 625.

The square root of this last product is obviously 5000; which, divide by 2, three times, or by 8, and we have 625 for the answer, or square root of the original sum.

7. What is the square root of 119025? 8. What is the square root of 75625?

Ans. 345.

Ans. 275.

(ART. 138.) ABBREVIATIONS IN CUBE ROOT.

1. What is the cube root of 91125?

Multiply by

Ans. 45.

8

729000

Now, 729 being the cube of 9, the root of 729000 is 90; divide this by 2, the cube root of 8, and we have 45, the answer.

2. The contents of a cubical cellar are 1953,125 cubic feet; what is the length of one of its sides?

The cube root of this is 50; divide by 4, because we multiplied by 8 twice, and we have 12,5, the answer.

When it is requisite to multiply several numbers together, and extract the cube root of their product, try to change them into cube factors, and extract the root before multiplication.

EXAMPLES.

1. What is the side of a cubical mound equal to one 288 feet long, 216 feet broad, and 48 feet high?

The common way of doing this, is to multiply these numbers together, and extract the root, a lengthy operation. But, observe, that 216 is a cube number, and 288 =2× 12× 12, and 48=4× 12; therefore, the whole product is 216× 8 × 12 × 12 X 12. Now, the cube root of 216 is 6, of 8 is 2, and of 123 is 12, and the product of 6X2X12=144, the answer.

2. Required the cube root of the product of 448×392, in a brief manner.

Ans. 56.

3. What is the side of a cubical mound, equal to one 144 feet long, 108 feet broad, and 24 feet deep?

Ans. 72.

4. A pile of wood is 160 feet long, 6 feet wide, and 9 feet high; what would be the side of a cubic pile containing the same quantity? Ans. 123√5=20,5+feet.

5. What shall be the side of a cubical cistern, to contain 64 hogsheads? Ans. 8,138 feet. Solution: 64 X 63 × 231=64×7×9×3×77=64.27.539 Cube root is 4×3×3√539=12×8,138 in., or 8,138 ft.

(ART. 139.) We can extract the root of cube numbers, by inspection, when they do not contain more than two periods.

EXAMPLES.

Find the cube root of 195112.

This number consists

of two periods; compare the superior period with the cubes in the table, and we find that 195 lies between 125 and 216. The cube root of the tens, then, must be 5. The unit figure of the given cube is 2, and no cube in the table has 2 for its unit figure, except 512, whose root is 8; therefore, 58 is the root required.

The number 912673 is a cube; what is its root?

Ans. 97.

Observe, the root of the superior period must be 9, and the root of the unit period must be some number which will give 3 for its unit figure when cubed, and 7 is the only figure that will answer.

Ans. 39.

Ans. 43.

Ans. 49.
Ans. 48.

The following numbers are cubes; required their roots. 1. What is the cube root of 59319? 2. What is the cube root of 79507? 3. What is the cube root of 117649? 4. What is the cube root of 110592? 5. What is the cube root of 357911? 6. What is the cube root of 389017? 7. What is the cube root of 571787?

Ans. 71.
Ans. 73.

Ans. 83.

When a cube has more than two periods, it can generally be reduced to two by dividing by some one or more of the cube numbers, unless the root is a prime number.

The number 4741632 is a cube; required its root. Here we observe, that the unit figure is 2; the unit figure of the root must, therefore, be the root of 512, as that is the only cube of the 9 digits whose unit figure is 2. The cube root of 512 is 8; therefore, 8 is the unit figure in the root, and the root is an even number, and can be divided by 2-and, of course, the cube itself can be divided by 8, the cube of 2.

8)4741632

592704

Y

Now, as the first number was a cube, and being divided by a cube, the number 592704 must be a cube, and, by inspection, as previously explained, its root must be 84, which, multiplied by 2, gives 168, the root required. The number 13312053 is a cube; what is its root?

Ans. 237.

As there are three periods, there must be three figures, units, tens, and hundreds, in the root; the hundreds must be 2, the units must be 7. Let us then find the second figure, or the tens, in the usual way, and we have 237 for the root.

Again, divide 13312053 by 27, and we have 493039 for another factor. The root of this last number must be 79, which, multiplied by 3, the cube root of 27, gives 237, as before.

The number 18609625 is a cube; what is its root? As this cube ends with 5, we will multiply it by 8:

18609625

8

148877000

As the first is a cube, this product must be a cube; and, as far as labor is concerned, it is the same as reduced to two periods, and the root, we perceive at once, must be 530, which, divided by 2, gives 265 for the root required.

TO FIND THE APPROXIMATE CUBE ROOT OF SURDS.

(ART 140.) The usual way of direct extraction, is too tedious to be much practiced, if any shorter method can possibly be obtained. By the invention of logarithms, a very short method has been found; but, before that event, several eminent mathematicians bestowed much time and labor to obtain short practical rules-and some of their rules are too ingenious and useful to be lost, notwithstanding the invention of logarithms has nearly superceded their absolute value in practice.

The following method is from Dr. Halley's algebraic formula, but more commodiously expressed; and after knowing the result of the analysis, we can apparently draw the same from mere observations on numbers, thus.

Let us take two cube numbers, say 125 and 216, whose roots are 5 and 6. There must be some law of comparison between them, either simple or complex. We evidently cannot make a proportional comparison between them, as 125 is not to 216, as 5 to 6. But let us double 125 and add 216, which gives 466, and double 216 and add 125, which gives 557. Now 466 is to 557 as 5 to 6, nearly. Here we have an approximate proportion. But to have a very near approximation, our cubes must have a nearer relative value; 216 is nearly double of 125: this should never be the case in making practical use of approximate proportions.

To show that we can be more accurate, observe that 216000 and 226981 are cubes; their roots are 60 and

61.

Now 216000 is not to 226981 as 60 to 61. But let us double the first and add it to the second, and double the second and add it to the first, and we shall have 658981 and 669962, which are to each other very nearly as 60 to 61.

Or, by the principles of proportion, the first is to the difference between the first and second, as is the third to the difference between the third and fourth. That is,

668981: 10981 :: 60 : 1 very nearly.

But 1 is the difference between the two roots, and if the last root, or 61, was unknown, this proportion would give it very nearly, by giving the difference between the two

roots.

EXAMPLES.

1. Required the cube root of 66.

The cube root of 64 is 4. Now it is manifest, that the cube root of 66 is a little more than 4, and by taking a similar proportion to the preceding, we have